Subgroups — C₂²×#16863434 #31719481

Order 17152.

3006 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524002	C₂	1	2	yes	maximal
2	C₂×#16863434	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523977	C₂	1	2	yes	maximal, complemented
4	C₂²×#14523975	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	—	2	12	no	—
7	C₂×#14524002	C₂²	1	24	yes	—
8	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
9	C₂²×C₂²⋊Dic₆₇	—	2	2	no	—
10	C₂²×C₄×Dic₆₇	—	2	2	no	—
11	#16863434	C₂²	1	64	yes	direct product factor
12	C₂×#14523977	C₂²	1	24	yes	complemented
13	C₂×#14523975	C₂²	1	24	yes	complemented
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₄	1	16	yes	complemented
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	12	yes	—
19	C₂³×Dic₆₇	—	2	4	no	—
20	#14524002	—	2	48	no	—
21	#14524002	C₂³	1	32	yes	—
22	C₂×C₄⋊Dic₆₇	—	2	12	no	—
23	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
24	C₂×C₂²⋊Dic₆₇	—	2	24	no	—
25	C₂×C₄×Dic₆₇	—	2	24	no	—
26	#14523977	C₂³	1	32	yes	complemented
27	#14523975	C₂³	1	32	yes	complemented
28	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₆₇	D₄	1	6	yes	—
31	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	96	yes	complemented
38	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	16	yes	—
41	C₂²×Dic₆₇	—	2	24	no	—
42	C₂²×Dic₆₇	—	4	16	no	—
43	C₄⋊Dic₆₇	—	2	24	no	—
44	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
45	C₂²⋊Dic₆₇	—	2	32	no	—
46	C₄×Dic₆₇	—	2	32	no	—
47	C₂⁴×C₆₇	—	2	8	no	—
48	C₂⁴×C₆₇	C₄⋊C₄	1	8	yes	—
49	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
50	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	48	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	64	yes	complemented
65	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
67	C₂³×C₆₇	—	2	24	no	—
68	C₂³×C₆₇	—	2	32	no	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂³×C₆₇	#135	1	8	yes	—
72	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
73	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
77	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
78	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
79	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
80	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
82	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
83	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
84	C₂×Dic₆₇	—	2	16	no	—
85	C₂×Dic₆₇	—	4	48	no	—
86	C₈×C₆₇	C₂³×C₄	1	8	yes	—
87	C₈×C₆₇	C₂³×C₄	1	8	yes	—
88	C₂²×C₆₇	—	2	48	no	—
89	C₂²×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	#427	1	16	yes	complemented
91	C₂²×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
92	C₂²×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
93	C₂²×C₆₇	C₂×#135	1	12	yes	—
94	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
95	C₄×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
96	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
97	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
98	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
99	Dic₆₇	—	4	16	no	—
100	C₂²×#427	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	8	no	—
102	C₂×C₆₇	C₂×#427	1	12	yes	complemented
103	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₆₇	C₂²×#135	1	2	yes	—
105	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
106	C₂²×#117	—	67	2	no	—
107	C₂×#427	—	67	24	no	—
108	C₂²×#107	—	67	2	no	—
109	C₂²×#106	—	67	2	no	—
110	C₆₇	C₂²×#427	1	1	yes	characteristic, complemented, Sylow
111	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
112	C₂³×C₈	Dic₆₇	1	16	yes	complemented
113	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
114	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₂³×C₈	C₂×D₆₇	1	12	yes	—
116	C₂×#117	—	67	24	no	—
117	C₂×#117	—	134	12	no	—
118	C₂²×C₄⋊C₄	—	67	4	no	—
119	C₂²×C₂²⋊C₄	—	134	2	no	—
120	C₂²×C₄²	—	134	2	no	—
121	#427	—	67	64	no	—
122	C₂×#107	—	67	24	no	—
123	C₂×#106	—	67	24	no	—
124	C₂⁵	D₂₆₈	1	1	yes	characteristic
125	C₂³×C₄	—	134	4	no	—
126	C₂³×C₄	Dic₁₃₄	1	8	yes	—
127	C₂³×C₄	D₂₆₈	1	6	yes	—
128	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
129	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
130	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
132	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
133	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
134	C₂²×C₈	C₂×Dic₆₇	1	96	yes	complemented
135	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
136	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
137	C₂²×C₈	C₂²×D₆₇	1	16	yes	—
138	#117	—	67	32	no	—
139	#117	—	134	48	no	—
140	C₂×C₄⋊C₄	—	67	24	no	—
141	C₂×C₄⋊C₄	—	134	12	no	—
142	C₂×C₂²⋊C₄	—	134	24	no	—
143	C₂×C₄²	—	134	24	no	—
144	#107	—	67	32	no	—
145	#106	—	67	32	no	—
146	C₂⁴	—	2	8	no	—
147	C₂⁴	C₄⋊Dic₆₇	1	8	yes	—
148	C₂⁴	C₂×D₂₆₈	1	6	yes	—
149	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
150	C₂²×C₄	—	268	16	no	—
151	C₂²×C₄	—	134	24	no	—
152	C₂²×C₄	C₄⋊Dic₆₇	1	48	yes	—
153	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
154	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
155	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
156	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
157	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
158	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
159	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
160	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
161	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
162	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
163	C₂×C₈	C₂²×Dic₆₇	1	24	yes	—
164	C₂×C₈	C₂²×Dic₆₇	1	24	yes	—
165	C₂×C₈	C₂²×Dic₆₇	1	64	yes	complemented
166	C₂×C₈	C₂³×D₆₇	1	4	yes	—
167	C₂×C₈	C₂³×D₆₇	1	4	yes	—
168	C₄⋊C₄	—	67	16	no	—
169	C₄⋊C₄	—	134	24	no	—
170	C₂²⋊C₄	—	134	32	no	—
171	C₄²	—	134	32	no	—
172	C₂³	—	2	24	no	—
173	C₂³	—	2	32	no	—
174	C₂³	C₂×C₄⋊Dic₆₇	1	24	yes	—
175	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
176	C₂³	#14524065	1	8	yes	—
177	C₂³	C₂²×D₂₆₈	1	4	yes	—
178	C₂³	C₂²×D₂₆₈	1	3	yes	—
179	C₂×C₄	—	268	48	no	—
180	C₂×C₄	—	134	16	no	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	48	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
184	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
185	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
186	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
187	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
188	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
189	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
190	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
191	C₈	C₂³×Dic₆₇	1	8	yes	—
192	C₈	C₂³×Dic₆₇	1	8	yes	—
193	C₂²	—	2	48	no	—
194	C₂²	—	2	8	no	—
195	C₂²	#16863434	1	16	yes	direct product factor
196	C₂²	C₂²×C₄⋊Dic₆₇	1	8	yes	—
197	C₂²	C₂²×C₄⋊Dic₆₇	1	6	yes	—
198	C₂²	C₂×#14524065	1	12	yes	—
199	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
200	C₄	—	268	16	no	—
201	C₄	C₂²×C₄⋊Dic₆₇	1	6	yes	—
202	C₄	C₂²×C₄⋊Dic₆₇	1	8	yes	—
203	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
204	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
205	C₂	—	2	8	no	—
206	C₂	C₂×#16863434	1	12	yes	direct product factor
207	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
208	C₂	C₂²×#14524065	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂×D₄
prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂²×#427
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#16863434	C₂	direct (N × H)
3. C₂²×#14523977	C₂	semidirect
4. C₂²×#14523975	C₂	semidirect
11. #16863434	C₂²	direct (N × H)
12. C₂×#14523977	C₂²	semidirect
13. C₂×#14523975	C₂²	semidirect
15. C₂³×C₈×C₆₇	C₄	semidirect
26. #14523977	C₂³	semidirect
27. #14523975	C₂³	semidirect
37. C₂²×C₈×C₆₇	C₂×C₄	semidirect
64. C₂×C₈×C₆₇	C₂²×C₄	semidirect
90. C₂²×C₆₇	#427	semidirect
102. C₂×C₆₇	C₂×#427	semidirect
110. C₆₇	C₂²×#427	semidirect
112. C₂³×C₈	Dic₆₇	semidirect
134. C₂²×C₈	C₂×Dic₆₇	semidirect
165. C₂×C₈	C₂²×Dic₆₇	semidirect
195. C₂²	#16863434	direct (N × H)
206. C₂	C₂×#16863434	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14524002	2
C₂×#16863434	2
C₂²×#14523977	2
C₂²×#14523975	2
C₂⁴×C₈×C₆₇	2
C₂²×#427	67

Subgroups — C22×#16863434 #31719481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863434 #31719481